In like manner it may be shown that the dihedral angles QB and QC are equal to the dihedral angles Q'B' and 'c' respectively.

Hence, § 494, if the equal angles are arranged in the same order, as in the first two figures, the two trihedral angles are equal; but if they are arranged in the reverse order, as in the first and third figures, the two trihedral angles are symmetrical.

Therefore, etc.

Q.E.D.

501. Cor. If two trihedral angles have three face angles of the one equal to three face angles of the other, then the dihedral angles of the one are respectively equal to the dihedral angles of the other.

SUPPLEMENTARY EXERCISES

Ex. 767. If a straight line is parallel to a plane, any plane perpendicular to the line is perpendicular to the plane.

Ex. 768. If a straight line intersects two parallel planes it makes equal angles with them.

Ex. 769. If a line is parallel to each of two planes, the intersections which any plane passing through it makes with the planes are parallel.

Ex. 770. The projections of parallel straight lines on any plane are either parallel or coincident.

Ex. 771. Find the locus of points which are equidistant from three given points not in the same straight line.

Ex. 772. From any point within the dihedral angle A-BC-D, EF and EG are drawn perpendicular to the faces AC and BD, respectively, and GH perpendicular to AC at H. Prove that FH is perpendicular to BC.

Ex. 773. If a plane is passed through the middle point of the common perpendicular to two straight lines in space, and parallel to both lines, it bisects every straight line drawn from any point in one line to any point in the other line.

Ex. 774. If the intersections of several planes are parallel, the perpendiculars drawn to them from any point lie in one plane.

Ex. 775. If two face angles of a trihedral are equal, the dihedral angles opposite them are also equal.

Ex. 776. A trihedral angle, having two of its dihedral angles equal, may be made to coincide with its symmetrical trihedral angle.

Ex. 777. In any trihedral the three planes bisecting the three dihedrals intersect in the same straight line.

Ex. 778. In any trihedral the planes which bisect the three face angles, and are perpendicular to those faces, respectively, intersect in the same straight line.

BOOK VIII

POLYHEDRONS

502. A solid bounded by planes is called a Polyhedron.

The intersections of the planes which bound a polyhedron are called its edges; the intersections of the edges are called its vertices; and the portions of the planes included by its edges are called its faces.

The line joining any two vertices of a polyhedron, not in the same face, is called a diagonal of the polyhedron.

503. A polyhedron having four faces is called a tetrahedron ; one having six faces is called a hexahedron; one having eight faces is called an octahedron; one having twelve faces is called a dodecahedron; one having twenty faces is called an icosahedron.

504. If the section made by any plane cutting a polyhedron is a convex polygon, the solid is called a Convex Polyhedron. Only convex polyhedrons are considered in this work.

PRISMS

505. A polyhedron two of whose faces are equal polygons, which lie in parallel planes and have their homologous sides parallel, and whose other faces are parallelograms, is called a Prism.

The two equal and parallel faces of the prism are called its bases; the other faces are called lat

eral faces; the intersections of the lateral faces are

called lateral edges; the sum of the lateral faces is called the lateral, or convex surface; and the sum of the areas of the lateral faces is called the lateral area of the prism.

The lateral edges of a prism are parallel and equal. § 153. The perpendicular distance between the bases of a prism is its altitude.

506. A prism is called triangular, quadrangular, hexagonal, etc., according as its bases are triangles, quadrilaterals, hexagons, etc.

507. A prism whose lateral edges are perpendicular to its bases is called a Right Prism.

508. A prism whose lateral edges are not perpendicular to its bases is called an Oblique Prism.

509. A right prism whose bases are regular polygons is called a Regular Prism.

510. A section of a prism made by a plane perpendicular to its lateral edges is called a Right Section.

511. The part of a prism included between one base and a section made by a plane oblique to that base, and cutting all the lateral edges, is called a Truncated Prism.

512. A prism whose bases are parallelograms is called a Parallelopiped.

513. A parallelopiped whose lateral edges are perpendicular to its bases is called a Right Parallelopiped.

514. A parallelopiped all six of whose faces are rectangles is called a Rectangular Parallelopiped.

515. A parallelopiped whose six faces are all squares is called a Cube.

516. The quantity of space inclosed by the surfaces which bound a solid is called the Volume of the solid.

A solid is measured by finding how many times it contains some other solid adopted as the unit of measure.

The units of measure for volume are the cubic inch, the cubic foot, the cubic yard, the cubic centimeter, the cubic decimeter, etc.

Suppose that the cube M is the unit of measure and that AB is the rectangular parallelopiped to be measured.

A

E

M

Apply an edge of M to each edge of AB and at the points of division pass planes respectively perpendicular to those edges. These planes divide AB into cubes, each equal to the unit M. It is evident that there will be as many layers of these cubes as the edge of M is contained times in the altitude of AB, that each layer will contain as many rows of cubes as the edge of M is contained times in the width of AB, and that each row will contain as many cubes as the edge of M is contained times in the length of AB; and, therefore, that the product of the numerical measures of the three dimensions of AB is equal to the number of times that M is contained in AB.

In this case the edge of M is contained 4 times in DE, 3 times in DA, and 5 times in DC; consequently, there are 5 cubes in each row, 3 rows in each layer, and 4 layers in the parallelopiped; that is, M is contained in AB 5 × 3 × 4 = 60 times, or the rectangular parallelopiped contains 60 cubic units.

Therefore, if the edge of M is a common unit of measure of the three dimensions of a rectangular parallelopiped, the product of the numerical measures of the three dimensions expresses the number of times that the rectangular parallelopiped contains the cube, and is the numerical measure of the volume of the rectangular parallelopiped.

517. For the sake of brevity, the product of the three dimensions is used instead of the product of the numerical measures of the three dimensions.

The product of three lines is, strictly speaking, an absurdity, but since the expression is used to denote the volume of a rectangular parallelopiped, it follows that the geometrical concept of the product of three lines is the rectangular parallelopiped whose edges they are.

Thus, DC DA X DE implies a product, which is a numerical result, but it must be interpreted geometrically to mean the rectangular parallelopiped whose edges are DC, DA, and DE.

SOLID GEOMETRY.-BOOK VIII.

reasons, the cube of a line must be interpreted geolean the cube constructed upon the line as an edge, the cube constructed upon a line may be indicated the line.

which have the same form are similar; those which volume are equivalent; and those which have the 1 volume are equal.

Proposition I

m * a prism. Since the faces are parallelograms, how does are with a rectangle having the same base and altitude? ateral edge as the base of each, how does the sum of the altiwith the perimeter of the right section? Since the lateral 1, how does the lateral surface of a prism compare with f its lateral edge and the perimeter of a right section? ectangle is the lateral surface of a right prism equivalent? The lateral surface of a prism is equivalent gle formed by a lateral edge and the perimet section.

epresentations of the solids referred to in this and the followaid the student very greatly in acquiring the correct geometSolids made from wood or glass may be procured, but it will the student to form them for himself from some plastic mate